
The first term is \(O_p(1)\)  by the noiseless extreme value theorem REF , case of \(\gamma <0\)  (recall that \(a_N = \theta _F-U_F(N)\)  by eq. (REF )). This term does not depend on \(T\) .

Second term is \(O_p(1)\)  by lemma REF . Lemma REF  applies to subsample \(s\) , since \((N, T)\)  satisfy conditions of proposition REF , and \(b=o(N)\) , hence \((b, T)\)  satisfy conditions of proposition REF  as well.

Last, \((\theta _F-U_F(t))\in RV_{\gamma }\)  by corollary 1.2.10 in [1]}. By proposition 0.5 in [2]} \( {(\theta _F- U_F(xt))}/{(\theta _F - U_F(t))}\rightarrow x^{\gamma }\)  uniformly on intervals of the form \((b, \infty )\) . Hence, using
\(b= N^m\) , \(m<1, \gamma <0\) , we obtain
\(\frac{\theta _F-U_F(N)}{\theta _F-U_F(N^m)} = \frac{\theta _F-U_F((N^{1-m}N^m))}{\theta _F-U_F(N^m)} \sim (N^{1-m})^{\gamma } \rightarrow 0.\) 
The last term is \(o(1)\) .

